On the Deformation of the Circumflex Coronary ArteryDuring Inflation Tests at Constant Length

U. Saravanan & S. Baek & K.R. Rajagopal &

J.D. Humphrey

Received: 7 February 2006 /Accepted: 4 May 2006 / Published online: 13 July 2006# Society for Experimental Mechanics 2006

Abstract Here we investigate whether the deforma-

tion observed in an experiment in which the porcine

circumflex coronary artery is subjected to inflation at

constant length included in the class, r¼rðRÞ, �¼Q,

z¼LZ. We find that this is not the case and discuss its

implications in the study of the mechanics of this

artery. Moreover, we identify and quantify the uncer-

tainty in the value of the invariants of the left Cauchy–

Green tensor inferred from the 2D motion of markers

affixed to the surface of the test specimen, and suggest

that 3D tracking of markers is needed due to inherent

bending and twisting induced by pressurization in

vitro.

Keywords Coronary artery . Inflation tests . Non-

axisymmetric deformation . 2D marker tracking .

Calibration

Introduction

The stress experienced by cells in the arterial wall

modulates the structural and functional characteristics

[1–5]. Hence, we must quantify their transmural stress

distribution so that the role of mechanical factors can

be understood in arterial physiology and pathophysi-

ology. Towards this end, the present study aims to

glean information regarding the deformation field of

the circumflex coronary artery inflated at constant

length in vitro.

Coronary artery disease is a leading cause of

mortality in the western world, yet few studies address

the mechanics of these arteries (see [6]). Most of the

available data [7–14] concern the realized outer

diameter, for a given radial component of the normal

stress at the inner surface of the artery (usually

referred to as the lumenal pressure), when length is

held fixed during in vitro experiments or in unknown

conditions1 in vivo. These measurements alone are

inadequate to characterize the inhomogeneity or

anisotropy of the artery, for many constitutive pre-

scriptions can reasonably fit the available experimental

data.2 Hence, there is a need for additional informa-

tion about the deformation field.

One can consider the possible inhomogeneity and

anisotropy by observing the deformation of markers

placed on the surface of the circumflex artery. From this,

we could obtain the circumferential and axial variations

of the surface deformation field. Of course, this too is

insufficient for obtaining robust 3D constitutive relations,

which would also require information about the trans-

Experimental Mechanics (2006) 46: 647–656

DOI 10.1007/s11340-006-9036-2

U. Saravanan (*)Department of Civil Engineering, IIT-Madras Chennai,Tamil Nadu 600036, Indiae-mail: saran@iitm.ac.in

S. Baek : J.D. HumphreyDepartment of Biomechanical Engineering,Texas A&M University,College Station TX-77845-3120, USA

K.R. RajagopalDepartment of Mechanical Engineering,Texas A&M University,College Station TX-77845-3123, USA

2 This is the case for other arteries as well. Also, while Laplace’slaw is reasonable to estimate the circumferential stresses forhomogeneous, isotropic, thin walled bodies, it is not forinhomogeneous, anisotropic bodies and for those bodies whichhave internal stresses even when free of traction on theboundary.

1 Perivascular tissue may allow small changes in length and limitthe amount of distension, due to the generation of a radialcomponent of the normal stress at the outer surface.

SEM

mural variation of the deformation field (to infer the

radial inhomogeneity). Further, one expects that anisot-

ropy and the stress field in the boundary traction free

reference configuration of the artery will be revealed by

studying the eigen-directions of the left Cauchy–Green

stretch tensor in conjunction with the estimated eigen-

directions of the Cauchy stress tensor, and inhomogeneity

by the non-uniformity of the principal invariants of the

right Cauchy–Green stretch tensor evaluated over theo-

retically predetermined surfaces. Herein we focus on

invariants rather than eigen-directions and address the

possible existence of inhomogeneity.

Of course, one can glean information about inho-

mogeneity by performing experiments on the intact

artery and repeating the same after separating the

media and adventitia, as done by Lu et al. [15] for

coronary arteries and by Von Maltzahn et al. [16] for

carotid arteries. Apart from difficulties in dissection,

this will alter the self equilibrating stress field in the

boundary traction free reference configuration of the

body. Hence, any observed changes in mechanical

response would be due in part to the altered stress field

in the configuration chosen as reference and different

material properties of the media and adventitia.

Therefore, there is an advantage to infer the stress

field in the reference configuration and inhomogeneity

within the context of a boundary value problem on

intact arteries rather than inferring them from destruc-

tive experiments. Since different stress fields in the

reference configuration and inhomogeneities would

theoretically result in different deformation fields, we

seek to deduce them by obtaining information regard-

ing the deformation. However, many distinct stress

fields in the reference configuration and inhomogene-

ities could result in deformation fields that are too

close to be resolved experimentally due to the

uncertainty associated with the estimated deformation

field, leading to non-uniqueness with regard to the

inverse problem at hand. Hence, we also study the

experimental uncertainty as well.

Here we illustrate our ideas by studying the

response of porcine circumflex coronary arteries sub-

jected to inflation at constant length. It is commonly

assumed that the deformation of arteries subjected in

vitro to inflation at constant length is given by

r ¼ rðRÞ; � ¼ Q; z ¼ LZ; ð1Þ

where ðR;Q;ZÞ denotes the cylindrical polar coordi-

nates of a typical material point in a reference

configuration and ðr; �; zÞ denotes the cylindrical polar

coordinates of a typical material point in the current

configuration. In general, such a deformation is

possible in annular right circular cylinders only when

the body is at most radially inhomogeneous and the

stress field in the reference configuration could vary at

most radially. Otherwise, the equilibrium equations

cannot be satisfied. This means that if the observed

deformation is of the form (1), then the two-layered

annular right circular cylinder approximation of the

artery can be accommodated. Here we investigate

whether a deformation of the form (1) is possible in a

circumflex coronary artery that is inflated and held at

constant length, in vitro. In vivo, it has been shown by

Pao et al. [17] that coronary arteries are subjected to

axial extension and torsion in addition to the longitu-

dinal shear at the inner surface of the artery due to the

flowing blood and inflation due to the blood pressure;

this complexity arises, in part, because of changes in

shape and size of the heart during the cardiac cycle. In

vitro we also find that the artery bends and twists on

inflating when holding it at constant length; this

implies that the deformation is not of the form (1). It

is pertinent to note that Draney et al. [18], studying

porcine aortic wall motion in vivo using cine phase

contrast MRI, report circumferentially non-uniform

deformation. Thus, it appears that in many arteries the

deformation might not be of the form (1), which has

profound implications in the study of the mechanics of

these arteries which we discuss in some detail.

Materials and Methods

Experimental System

A computer controlled system, originally designed and

built to test embryonic chick hearts subjected to low

pressures [19], was adapted to test porcine circumflex

coronary arteries. The overall system consists of three

main subsystems (Fig. 1).

We employ a video-based system that allows 2D

tracking of up to 12 surface markers. This system

consists of a microscope (Olympus SZ60) with an

auxiliary viewing port (SZ-PT), a charged couple

device (CCD) camera (Javelin JE-7442), a VCR (Sony

SVT-S3100), two B&W monitors (Sony SSM-171 for

specimen preparation and a Panasonic TR-930B for

visualizing the on-line tracking of fiducial markers),

and a video frame grabber board (Data Translation

DT-2853SQ) that captures 8 bit gray scale images as

512� 512 pixel arrays. Markers are tracked online at

30 Hz using the correlation-based algorithm reported

in Downs et al. [20]. To maintain the focus of the

markers on the surface of the artery during Bextreme^deformations, a manually controlled focussing mecha-

nism translates the microscope optics vertically. This

648 Exp Mech (2006) 46: 647–656

SEM

was achieved by replacing the rack and pinion

microscope stand with a motorized vertical translation

stage (Newport Corporation 426 and CMA-25CC 861

controller) mounted on a damped mounting rod

(Thorlabs DP14).

Second, the cannulated specimen is held and loaded

by means of a mounting system, so that the artery is

submerged in the test chamber containing a physio-

logic solution. The specimen can be axially stretched

using paired computer controlled actuators (Newport

Corp.) through precision x–y–z stages that adjust the

position of the specimen within the video field of view.

The x–y–z stages also maintain the length of the artery

during inflation. One end of the specimen is attached

to a computer controlled syringe pump (World Preci-

sion Instruments SP210iw) and the other end to a

pressure transducer (Sensotec) and load cell (Senso-

tec), as shown in Fig. 1. The syringe pump, fitted with a

3 cc syringe (Hamilton 1705TLL), is controlled by the

computer via ASCII commands that allows cyclic

pressurization tests over a wide range of infusion rates.

The pressure transducer has a sensing range from 0 to

258 mmHg, with a factory reported accuracy of

0.26 mmHg. The axial load cell has a sensing range

from 0 to 250 g with a reported accuracy of 0.25 g. The

pressure transducer and the load cell are sampled via a

12-bit analog to digital (A/D) board (Data Translation

DT2831) in the computer.

Third, the experiments and data collection (video,

axial load and lumenal pressure) are controlled by

keyboard commands via a custom C code running on

an Intel Pentium II computer (Compaq Deskpro,

RAM reduced to 8 MB to accommodate the online

tracking algorithm). The stored data are analyzed

offline using a custom matlab code.

The pressure transducer was calibrated using a

sphygmomanometer and the load cell using standard

weights. The video system was not calibrated because

we were interested only in the dimensionless gradient

of the deformation.

Computation of the Deformation Gradient

We estimate a smooth deformation field from posi-

tions of Fn_ markers in the reference and current

configuration and then compute the gradient of the

deformation field. In general, it is not possible to

experimentally determine the exact deformation field

by tracking Fn_ markers, we can at most get a reasonable

approximation of the deformation field and its gradient

locally. However, we can verify if the observed defor-

mation field is consistent with that theoretically pre-

dicted. Towards this end we shall see that if the artery

were to deform as given in equation (1), then it suffices

to assume that the deformation field on the surface of

the specimen in a centered gage length is given by

x ¼ ax1X þ ax

2Z þ ax3; z ¼ az

1X þ az2Z þ az

3; ð2Þ

where axi and az

i are constants and ðx; zÞ denotes the

2D coordinates of the markers in the current configu-

ration and ðX;ZÞ the corresponding coordinates in a

reference configuration. Knowing the location of three

markers in the current and reference configurations,

we can obtain ðaxi Þ

c and ðazi Þ

c, with the value of axi and

Fig. 1 Schematic of theexperimental setup

Exp Mech (2006) 46: 647–656 649

SEM

azi corresponding to the cth choice of three markers,

from solving the linear equations

X1 Z1 1

X2 Z2 1

X3 Z3 1

0B@

1CAðax

1Þc

ðax2Þ

c

ðax3Þ

c

8><>:

9>=>;¼

x1

x2

x3

8><>:

9>=>;;

X1 Z1 1

X2 Z2 1

X3 Z3 1

0B@

1CAðaz

1Þc

ðaz2Þ

c

ðaz3Þ

c

8><>:

9>=>;¼

z1

z2

z3

8><>:

9>=>;;

ð3Þ

where Xi and Zi are coordinates in the reference

configuration of the ith marker that belongs to the

three markers that have been selected and xi and zi are

the coordinates of the same ith marker in the current

configuration. It is then straightforward to see that the

invariants, trðC2DÞ and detðC2DÞ, are given by

ðI2D1 Þ

c ¼ ðax1Þ

c2 þ ðax2Þ

c2 þ ðaz1Þ

c2 þ ðaz2Þ

c2;

ðI2D3 Þ

c ¼ ½ðax1Þ

cðaz2Þ

c � ðax2Þ

cðaz1Þ

c�2;ð4Þ

respectively, where, C¼FtF is the right Cauchy-Green

stretch tensor and F is the deformation gradient. Thus,

we find ðI2D1 Þ

c and ðI2D3 Þ

c corresponding to the cth

choice of three markers. Figure 2 shows a typical

selection of 12 sets of three markers where the markers

are at the vertices of each triangle and are denoted by

arabic numerals. Here we choose different sets of

three markers such that the triangles formed with

these markers as vertices have no overlapping areas.

Sometimes, one or two of the markers would wash off,

in which case the triangles were formed with remaining

markers still ensuring no overlap of the areas.

A study on the quality of approximation

Next, let us study in some detail the consequences of

approximating the deformation (1) using a linear

polynomial, (2). Let ðRo;Q1;Z1Þ, ðRo;Q2;Z2Þ, ðRo;Q3;

Z3Þ denote the cylindrical polar coordinates of three

markers in the reference configuration and ðro; �1; z1Þ,ðro; �2; z2Þ, ðro; �3; z3Þ the cylindrical polar coordinates

of the same markers in the current configuration. Now,

equation (3) becomes

Ro cosðQ1Þ þ xo Z1 1

Ro cosðQ2Þ þ xo Z2 1

Ro cosðQ3Þ þ xo Z3 1

0B@

1CA

ax1

ax2

ax3

8><>:

9>=>;¼

ro cosð�1Þ þ xo

ro cosð�2Þ þ xo

ro cosð�3Þ þ xo

8><>:

9>=>;;

Ro cosðQ1Þ þ xo Z1 1

Ro cosðQ2Þ þ xo Z2 1

Ro cosðQ3Þ þ xo Z3 1

0B@

1CA

az1

az2

az3

8><>:

9>=>;¼

z1

z2

z3

8><>:

9>=>;;

ð5Þ

where ro¼rðRoÞ and xo shifts the origin from the axis

of the annular cylinder to some point outside the

cylinder, which happens when local coordinates of the

frame grabber board are used. Solving the above

equations for axi and az

i and using �¼Q and z¼LZ,

we obtain ax1¼ro=Ro, ax

2¼0, ax3¼ðRo � roÞxo=Ro, az

1¼az

3¼0 and az2¼L. Thus, the value of the constants a

ji are

independent of the choice of the three markers selected,

as long as they are on the surface of the artery. The

values of

I2D1 ¼ ro

Ro

� �2

þL2 and I2D3 ¼ ro

RoL

� �2

; ð6Þ

are constant when inferred from tracking markers on

the surface of the artery, if assumed to be a right

circular annular cylinder and the actual deformation is

given by equation (1). It is this constancy of the

principal invariants that we propose to test. Also, we

have verified that equation (2) is sufficient for com-

puting the deformation gradient when the actual

deformation is given by equation (1).

Sample Preparation

Animal care in this study conformed to the guidelines

of the University Laboratory Animal Care Committee,

Texas A&M University. Hearts of micro-mini pigs

(Panapinto Micro Minipigs; Mansonville, CO) used in

another (approved) study were harvested and trans-

ported in ice-cold normal saline. Two to four cm long

segments of the circumflex artery, from its origin at the

left coronary artery, were dissected (see Fig. 3) and its

side branches were ligated using 2-0 or 3-0 nylon

150 200 250 300 350 400

0

50

100

150

200

250

Z (Pixel)

X (

Pix

el) I

II

III

IV

V

VI

VII

VIII

IX X

XI XII

1

2

3

4

5

6

7

8

9

10

11

12

Fig. 2 Selection of triangles (I–XII) in the reference configura-tion to compute the deformation field by tracking 12 markers (1–12) using equation (2)

650 Exp Mech (2006) 46: 647–656

SEM

braided sutures, depending on the size of the branches.

Care was exercised to ensure that minimal perivascular

tissue was attached to the ligatures (see Fig. 4). The

excised circumflex artery was mounted on two stainless

steel cannulae, with outer diameters nearly equaling

the inner diameter of the circumflex artery and care

being exercised to ensure that the artery did not twist.

Then, the vessel was held fixed at its unloaded length,

denoted Lo. It should be pointed that the in vivo

length of the circumflex changes with the inflation and

deflation of the heart and hence is not well defined.

Subsequently 100 mm diameter black spheres (Interac-

tive Medical Technologies) made of polystyrene

divinylbenzene were placed on the surface of the

artery using fine tipped forceps and glued (Perma-

bond) using a glass micropipet (WPI PG52150-4)

which was formed using a pipet puller (World Preci-

sion Instruments PUL-1) and by grinding it down to

the desired diameter of 50 mm. The location of the

markers was approximately at the center of the artery

and not near the ligations, when possible.

Protocols

On each of 30 specimens the following protocols were

performed:

1. The locations of the markers on the adventitia, in a

configuration free of boundary traction, were

recorded under quiescent conditions and defined

the native reference configuration. The invariants

I2D1 and I2D

3 , were computed using this as the

reference configuration.

2. The specimen was translated along the x direction

using the actuators in the precision x–y–z stages

and the motion of the markers recorded. This

protocol was performed to estimate the error in the

video tracking system.

3. The microscope was translated vertically using the

motorized actuator described before, and the

motion of the markers tracked. This protocol was

performed to estimate the error in the video

tracking system due to changing focus.

4. The artery was cyclically inflated and deflated from

1 mm Hg to 120 mm Hg for five cycles at constant

length in a normal saline bath (154 mM NaCl) by

infusing and withdrawing normal saline at 15ml=s.

The response was called native because the artery

was assumed to be in the native state i.e., isolated

from natural or artificial hormonal and neural

stimuli.

Other protocols were performed in many speci-

mens, including tests in solutions with high potassium

concentration to activate the smooth muscle followed

by tests in low calcium solutions to deactivate the

smooth muscle. Because the key finding on inhomo-

geneity was observed independent of the test solution,

the normal saline results are given here. See Saravanan

[21] for further details and results.

Results

Uncertainty Analysis

Any experimental measurement has some uncertainty

associated with it and it is necessary to estimate the

level of uncertainty. Towards this end, we start with

examining the error associated with determining the

location of the centroid of each marker. That is, the

determined location of a marker will usually be within

only a certain number of pixels of the actual location

thereby inducing an error. Further, the location of a

marker can only change discretely during tracking, as

opposed to continuous variations in reality. To get an

estimate of these errors, three protocols elaborated

above were used.

First, the initial location of markers was recorded

over 10 seconds in the reference configuration. The

Fig. 3 Schematic of the location from where the circumflexcoronary artery is dissected. LAD stands for left anteriordescending artery

Fig. 4 A dissected circumflex coronary artery. Note the naturalcurvatures

Exp Mech (2006) 46: 647–656 651

SEM

mean of all the repeated recordings was taken as the

location of a marker. Now, for the locations of the

markers in the current configuration, we used the same

readings obtained while recording the initial location

of the markers. Hence, ideally ðI2D1 Þ

c¼ 2 and ðI2D3 Þ

c¼ 1

and any deviation from these values signifies an error.

The deviation occurs because the centroids are deter-

mined within an accuracy of �a pixels due to inherent

noise in the system. When the deformation was

approximated by a linear polynomial (2), which was

determined using three markers, ðI2D1 Þ

c�2 and

ðI2D3 Þ

c�1 varied by as much as �0:02. This error was

sensitive to the number of readings, Fb_, over which the

averaged location of the markers was computed in the

current configuration, the polynomial used to approx-

imate the deformation, and the particular selection of

the markers. This error decreased as the number of

readings over which the marker location was averaged

increased, and decreased as the area subtended by the

triplets increased. However, as the area subtended by

the markers increased, the quality of the approximation

of the deformation and its gradient by linear polyno-

mial decreased. Hence, a balance has to be maintained.

Second, the repeated recording of an initial static

configuration of the markers was used to find the

location of the markers in the reference configuration

and protocol 2, in which the markers were translated

as a rigid body along the x direction, was used for the

location of the markers in the current configuration.

As before ideally, ðI2D1 Þ

c¼ 2, ðI2D3 Þ

c¼ 1. A represen-

tative plot of these errors for ðI2D1 Þ

c � 2 is shown in

Fig. 5(a) when the deformation was approximated by a

linear polynomial, determined using three markers,

and when the markers were manually translated by 300

pixels along the x direction in 18 seconds.3 Figure 5(b)

plots the histogram from the data in Fig. 5(a). Similar

plots were obtained for ðI2D3 Þ

c � 1 (not shown). While

the error increased with decreases in the area sub-

tended by the triplet of three markers, it initially

decreased but then increased with increasing number

of readings over which the averaged location of the

markers was computed. Hence, there exists an optimal

number of readings averaging over which the error is a

minimum. From this study, we found that averaging

over ten readings of the marker location appears to be

optimal in finding the true mean.

Finally, the third protocol was used to estimate the

error introduced due to changing focus. The reference

configuration was inferred as before, but the current

configuration of the markers was inferred from record-

ings in protocol-3. Again, ideally ðI2D1 Þ

c¼ 2 and

ðI2D3 Þ

c¼ 1. When the deformation was approximated

by a linear polynomial, using three markers, and when

the microscope was translated by �1250mm, the

estimated value of ðI2D1 Þ

c�2 and ðI2D3 Þ

c�1 changed by

as much as �0:06. This error was sensitive to the same

variables described above and changed in the same

manner as illustrated in Fig. 5. From these studies it

was concluded that the error in the estimated invari-

ants is �0:06.

Next, we examined the gross error associated with

2D tracking of the markers. First, we note that

markers placed on the surface of an artery do not lie

on a plane. Further, if the deformation is not of the

form (1), significant systematic errors can be intro-

duced due to the limitation of 2D tracking. To

estimate the error, a numerical simulation was per-

0 50 100 150

−0.05

0

0.05

Non dimensional time

(I12D

)c 2

(a)

0.05 0 0.050

100

200

300

400

500

µ = − 0.003

σ = 0.017

(I12D)c − 2

Fre

quen

cy

(b)

−

1 2 42 4 52 3 53 5 64 5 75 7 85 6 86 8 97 8 108 10 11

− − −−−−

− − −−−−

− − −−−−

− − −−−−

− − −

8 9 11− − −9 11 12− − −

− − −

Fig. 5 Plot of ðI2D1 Þ

c�2 as a (a) time series (b) histogram while

translating the artery along the x direction as a rigid body. Each

symbol type in panel (a) denotes a different triplet of markers

3 Typically, while inflating the artery the markers translate 300pixels along the x direction in 32 seconds

652 Exp Mech (2006) 46: 647–656

SEM

formed. For a rigid body rotation about its axis, i.e.,

the artery was assumed to deform as

r ¼ R; � ¼ � þ Q; z ¼ Z; ð7Þ

and the value of the invariants was computed theoret-

ically under the assumption that the markers were

tracked in 2D using equation (5). Figure 6 reveals that

the value of the invariants are far from their expected

values, ðI2D1 Þ

c¼ 2 and ðI2D3 Þ

c¼ 1. Thus, while this ex-

perimental set-up is ideal if the artery deforms as given

by equation (1), caution has to be exercised on inter-

preting the data when the deformation differs.

Next, we turn our focus to errors that are more

difficult to quantify. Some twist could be introduced

inadvertently while mounting the artery, and the effect

of this on the deformation of the artery while inflating

requires quantification. Towards this end, holding one

end fixed, the other end of the artery was rotated

manually so that the twist per unit length W was

approximately 1 deg/mm. Figure 7 depicts ðI2D1 Þ

c for

combined inflation at a fixed extension and twist of the

artery; note that the variation of ðI2D3 Þ

c is similar to

that of ðI2D1 Þ

c. In the same figures the response of the

artery when there was no twist is presented for

comparison. The presence of twist not only increased

the value of the invariants for a given pressure, as it

should, the variation was also larger when evaluated at

different locations on the surface of the artery. This

dependence of the invariants on the location on the

outer surface at which they were evaluated is expected

because twisting causes out of plane deformation of

the markers, which the 2D tracking system was unable

to capture correctly. Figure 8 depicts the invariants

when the artery was inflated by applying a pressure at

its inner surface from the twisted reference configura-

tion. In other words, the only difference between

Fig. 8 and Fig. 7(b) is the reference configuration used

to compute the value of the invariants. This is the

situation when the artery is mounted incorrectly, i.e.,

with a twist. Comparing Fig. 8 with Fig. 7(a), we find

that while the magnitude of the invariants were com-

parable when one accounts for the axial extension, their

variation with the choice of the markers was different

for lower pressures. This is because the invariants are

constrained to be ðI2D1 Þ

c¼ 2 and ðI2D3 Þ

c¼ 1 in the latter

case when the pressure is 0.

Finally, gluing markers onto the surface of the

artery introduces some error in the measured defor-

mation. The motion of the markers would also be

Fig. 6 Theoretically computed plots of I2D1 �2 and I2D

3 �1 when

the deformation of the artery is given by equation (7) for various

values of �. Note that both I2D1 �2 and I2D

3 �1 are close to each

other. The X and Z values listed in the figure gives the initial

coordinates of the markers constituting the triplet

0 20 400

0.5

1

1.5

2

β (deg)

Inva

riant

s

X = [157 152 110]

Z = [200 160 194]

(I12D)c − 2

(I32D)c − 1

0 20 40 60 80 100 120

0

0.5

1

1.5

Pressure (mm Hg)

(I12D

)c − 2

1 2 42 4 52 3 53 5 64 5 75 7 85 6 86 8 117 8 98 9 108 10 11

(a)

0 20 40 60 80 100 120

0

0.5

1

1.5

Pressure (mm Hg)

(I12D

)c 2

(b)

−

− − −−−−

− − −−−−

− − −−−−

− − −−−−

− − −−−−

− − −

Fig. 7 Plot of ðI2D1 Þ

c�2 when L¼1:1aW¼0bW¼1deg=mm for a

29 mm long vessel and inflating the artery to 120 mm Hg

Exp Mech (2006) 46: 647–656 653

SEM

influenced by the ligations of the side branches. These

errors are not quantified here, as they are difficult to

assess. However, these can be viewed as perturbations

that are a necessary adjunct of the experimental

measurements.

Results from Inflation Tests

Figure 9 displays how the pressure varied with the

invariants computed with respect to the native refer-

ence configuration corresponding to the 4th cycle of

loading and unloading. Contrary to expectation, we

see from the figure that the invariants seem to vary

spatially. The spatial variation of the invariants is far

more than can be accounted for by the error in the

location of their centroid, which was estimated to

cause a deviation of �0:06 in the value of the

invariants. Hence, the deformation of the artery

subjected to inflation at constant length is not given

by equation (1). The same conclusion was reached for

all the specimens tested. Although not shown, the

variability in the value of the invariants depended on

the vascular smooth muscle tone and the axial stretch

ratio at which the vessel was held constant (see

Saravanan [21]). In fact, during the experiments one

could observe bending and twisting of the artery when

inflated, at a constant length, which we hypothesize to

be responsible for the observed variation in the value

of the invariants with the choice of the markers.

It is pertinent to note that Han and Fung [22], while

investigating the pre-strain in porcine aorta by intro-

ducing a radial cut on a transverse section, also report

local circumferential variation of the components of

the pre-strain apart from the radial variation (see their

Fig. 3), that is for small variations in the circumferen-

tial and radial location. However, globally the circum-

ferential variation was not statistically significant.4 In a

pilot study, we found that the deformation field varied

significantly for diametrically opposite locations during

inflation of the artery at constant length (see Fig. 10).

These observations suggest further that the deforma-

tion of the circumflex coronary artery inflated at

constant length does not correspond to equation (1).

Fig. 9 Plot of ðI2D1 Þ

c�2 for (a) specimen-1 (b) specimen-2 com-

puted for various marker sets while inflating the porcine circumflex

artery, in the native state at constant length, Lo. A similar plot was

obtained for ðI2D3 Þ

c�1 in the case of both specimens (not shown).

Symbols are the same as in Fig. 7

0 20 40 60 80 100 120

0

0.5

1

1.5

Pressure (mm Hg)

(I12D

)c 2

(a)

0 20 40 60 80 100 120

0

0.5

1

1.5

Pressure (mm Hg)

(I12D

)c 2

(b)

−−

0 20 40 60 80 100 120

0

0.5

1

1.5

Pressure (mm Hg)

(I12D

)c − 2

Fig. 8 Plot of ðI2D1 Þ

c�2 when L¼1, W¼1deg=mm and inflating

the artery to 120 mm Hg. A similar plot was obtained for ðI2D3 Þ

c�1(not shown). Symbols are the same as in Fig. 7

4 This just means that the deformation is a non-linear function ofthe radial location contrary to the assumption made whilecomputing the strain from the location of the markers in thecurrent and reference configuration. In fact, the theoreticallydetermined deformation field, as reported by them, is a non-linear function of FR_, the radial location.

654 Exp Mech (2006) 46: 647–656

SEM

Discussion

Unlike the common carotid or basilar artery, the

excised circumflex coronary artery is not straight (see

Fig. 4): it has a natural curvature and it tapers, in

addition to having many branches that need to be

ligated. These ligations and the geometry of the vessel

can result in the deformation not being of the form (1).

Fosdick [23] has shown that the deformation of the

form5

r ¼ f ðR; tÞ; � ¼ Q; z ¼ �Z; ð8Þ

is possible in all homogeneous, incompressible, isotro-

pic bodies when inertial effects are taken into account.

This result can be extended to include all radially

inhomogeneous, incompressible, isotropic bodies. For

certain classes of stored energy function that seem to

describe arteries reasonably well, Humphrey and Na

[24] have shown that the difference between the quasi-

static and the dynamic solution is small when the rate

of change of the radial component of the normal stress

at the inner surface coincides with the observed

variation of the blood pressure. In our study, the rate

of change of the radial component of the normal stress

was much smaller6 than that of the rate of change in

the blood pressure. Hence, it seems unlikely that the

observed difference in the deformation could be

explained by including inertial effects.

We found that when a small segment of the

circumflex coronary artery obtained from the proximal

portion (i.e., near the aorta) was cut radially, the

opening angle (as defined in Choung and Fung [25])

was greater than that obtained from the distal portion,

i.e., the opening angle changes significantly along the

axial direction over a 20 to 40 mm length of the

circumflex coronary artery.7 This empirical observa-

tion suggests that the stress field in the reference

configuration varies axially as well as radially.

An anisotropic body cannot support an arbitrary

stress field in the reference configuration. Coleman

and Noll [27] determined the states of stresses that are

compatible with the anisotropy of the body, within the

context of elasticity. Hoger [28] determined restric-

tions on the residually stressed field that, in addition to

being consistent with the anisotropy of the body,

satisfies the equilibrium equations and the appropriate

traction boundary condition. The fact that the body

under consideration is anisotropic and inhomogeneous

and has to satisfy the equations of equilibrium implies

that the artery ought to bend and twist when subjected

to the type of loadings that were considered in this

paper. Of course, either the anisotropy or inhomoge-

neity or both might be responsible for the observed

deformation.

It has been conjectured that bending and twisting

are responsible for coronary arteries being prone to

atherosclerosis [29–31]. Hence, it becomes necessary to

investigate the cause for bending and twisting of the

artery. This would at the least require the capability to

track the surface markers in 3D, to measure the torque

that needs to be applied while inflating the vessel to

maintain a given angle of twist, and to investigate the

orientation of the collagen, elastin, and vascular

smooth muscle cells. Tracking surface markers in 3D

can be accomplished with two cameras (i.e., a biplane

video system) and the torque can be measured with a

standard transducer. Although, pressure diameter

studies provide some information we conclude, that

future Binflation-extension^ experiments on arteries

having natural out of plane curvatures be augmented

with these experimental capabilities and that histology

be performed regionally as a function of axial and

circumferential location, not just radial.

Acknowledgments This work was supported, in part, by NIHgrant R01-HL64372. We thank Dr. J.J. Hu for the artwork of theexperimental setup.

5 ðR;Q;ZÞ are the cylindrical polar coordinates of a typical pointin the reference configuration and ðr; �; zÞ are the cylindricalpolar coordinates of a typical point in the current configuration.6 In the present case, the pressure increased approximately atthe rate of 1 mmHg/s. In the experiments we maintained theinfusion and withdrawal rate to be constant and not the pressure. 7 The same was shown for rat aorta by Liu and Fung [26].

0 20 40 60 80 100 120

0

0.5

1

1.5

Pressure (mm Hg)

(I12D

) c −

2Std. Err at loc 1Mean at loc 1Std. Err. at loc 2Mean at loc 2

−−

−−

Fig. 10 Plot of ðI2D1 Þ

c�2 inferred from diametrically opposite

locations while inflating the circumflex artery holding it at a constant

length, 1:1Lo. A similar plot was obtained for ðI2D3 Þ

c�1 (not shown)

Exp Mech (2006) 46: 647–656 655

SEM

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